SIGNALS AND SYSTEM

SURAJ MISHRA

SUMIT SINGH

AMIT GUPTA

PRATYUSH SINGH

(E.C 2ND YEAR ,MCSCET) 1

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Topics

Introduction Classification of Signals Some Useful Signal Operations Some useful signal models

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Introduction

The concepts of signals and systems arise in a wide variety of areas:communications, circuit design, biomedical engineering, power systems, speech processing, etc.

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What is a Signal?

SIGNAL A set of information or data. Function of one or more

independent variables. Contains information about the

behavior or nature of some phenomenon.

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Examples of Signals

BRAIN WAVE

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Examples of Signals

Stock Market data as signal (time series)

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What is a System?

SYSTEMSignals may be processed further

by systems, which may modify them or extract additional from them.

A system is an entity that processes a set of signals (inputs) to yield another set of signals (outputs).

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What is a System? (2)

A system may be made up of physical components, as in electrical or mechanical systems (hardware realization).

A system may be an algorithm that computes an outputs from an inputs signal (software realization).

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Examples of signals and systems

Voltage (x1) and current (x2) as functions of time in an electrical circuit are examples of signals.

A circuit is itself an example of a system (T), which responds to applied voltages and currents.

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Some Useful Signal Some Useful Signal ModelsModels

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Signal Models: Unit Step Function

Continuous-Time unit step function, u(t):

u(t) is used to start a signal, f(t) at t=0 f(t) has a value of ZERO for t <0

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Signal Models: Unit Impulse Function

A possible approximation to a unit impulse:An overall area that has been maintained at unity.

Multiplication of a function by an Impulse?

b(t) = 0; for all t0is an impulse function which the area is b.

Graphically, it is represented by an arrow "pointing to infinity" at t=0 with its length equal to its area.

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Signal Models: Unit Impulse Function (3)

May use functions other than a rectangular pulse. Here are three example functions:

Note that the area under the pulse function must be unity.

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Signal Models: Unit Ramp Function

Unit ramp function is defined by: r(t) = tu(t)

Where can it be used?

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Signal Models: Exponential Function est

Most important function in SNS where s is complex in general, s = +j

Therefore,est = e(+j)t = etejt = et(cost + jsint)(Euler’s formula: ejt = cost + jsint)

If s = -j, est = e(-j)t = ete-jt = et(cost - jsint)

From the above, etcost = ½(est +e-st )

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Signal Models: Exponential Function est (2)

Variable s is complex frequency. est = e(+j)t = etejt = et(cost + jsint)

est = e(-j)t = ete-jt = et(cost - jsint)etcost = ½(est +e-st )

There are special cases of est :1. A constant k = ke0t (s=0 =0,=0)

2. A monotonic exponential et (=0, s=)

3. A sinusoid cost (=0, s=j)

4. An exponentially varying sinusoid etcost (s= j)

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Signals Classification

Signals may be classified into: 1. Continuous-time and Discrete-time signals 2. Deterministic and Stochastic Signal 3. Periodic and Aperiodic signals 4. Even and Odd signals 5. Energy and Power signals

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Continuous v/S Discrete Signals

Continuous-timeA signal that is specified for everyvalue of time t.

Discrete-timeA signal that is specified only at discrete valuesof time t.

Deterministic v/s Stochastic Signal

Signals that can be written in any mathematical expression are called deterministic signal.

(sine,cosine..etc) Signals that cann’t be written in mathematical

expression are called stochastic signals. (impulse,noise..etc)

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Periodic v/s Aperiodic Signals

Signals that repeat itself at a proper interval of time are called periodic signals.

Continuous-time signals are said to be periodic.

Signals that will never repeat themselves,and get over in limited time are called aperiodic or non-periodic signals.

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Even v/s Odd Signals

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Even v/s Odd Signals

A signal x(t) or x[n] is referred to as an even signal if CT: DT:

A signal x(t) or x[n] is referred to as an odd signal if CT: DT:

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Even and Odd Functions: Properties

Property:

Area: Even signal:

Odd signal:

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Even and Odd Components of a Signal (1)

Every signal f(t) can be expressed as a sum of even and odd components because

Example, f(t) = e-atu(t)

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Signal with finite energy (zero power)

Signal with finite power (infinite energy)

Signals that satisfy neither property are referred as neither energy nor power signals

Energy v/s Power Signals

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Size of a Signal, Energy (Joules)

Measured by signal energy Ex:

Generalize for a complex valued signal to: CT: DT:

Energy must be finite, which means

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Size of a Signal, Power (Watts)

If amplitude of x(t) does not → 0 when t → ∞, need to measure power Px instead:

Again, generalize for a complex valued signal to: CT:

DT:

OPERATIONS ON SIGNALS

It includes the transformation of independent variables.

It is performed in both continuous and discrete time signals.

Operations that are performed are-

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1.ADDITION &SUBSTRACTION

Let two signals x(t) and y(t) are given, Their addition will be,

z(t) = x(t) + y(t)

Their substraction will be,

z(t) = x(t) – y(t)

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2.MULTIPLICATION OF SIGNAL BY A CONSTANT

If a constant ‘A’ is given with a signal x(t)

z(t) = A.x(t)

If A>1,it is an amplified signal. If A<1,it is an attenuated signal.

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3.MULTIPLICATION OF TWO SIGNALS

If two signals x(t) and y(t) are given,than their multiplication will be

z(t) = x(t).y(t)

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4.SHIFTING IN TIME

Let a signal x(t),than the signal x(t-T) represented a delayed version of x(t),which is delayed by T sec.

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Signal Operations: Time Shifting

Shifting of a signal in time adding or subtracting the amount of the

shift to the time variable in the function. x(t) x(t–to)

to > 0 (to is positive value),signal is shifted to the right (delay).

to < 0 (to is negative value),signal is shifted to the left (advance).

x(t–2)? x(t) is delayed by 2 seconds. x(t+2)? x(t) is advanced by 2 seconds.

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Signal Operations: Time Shifting (2)

Subtracting a fixed amount from the time variable will shift the signal to the right that amount.

Adding to the time variable will shift the signal to the left.

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Signal Operations: Time Shifting

Shifting of a signal in time

5.COMPRESSION/EXPANSION OF SIGNALS

This is also known as ‘Time Scaling’ process. Let a signal x(t) is given,we will examine as

x(at)

where a =real number and how it is related to x(t) ?

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Time Scaling

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Signal Operations: Time Inversion

Reversal of the time axis, or folding/flipping the signal (mirror image) over the y-axis.

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